Finite Sample Analysis in Quantum Estimation

Book Preface

Recent dramatic development of theory and experiment for microscopic systems has made it possible to investigate Nature more deeply and to utilize Her often surprising properties for our own purposes. Tests of quantum entanglement is a good early example. In 1935 Einstein, Podolsky, Rosen, and Schrödinger pointed out that quantum mechanics allows the existence of a counterintuitive correlation, now called entanglement, in quantum systems [1, 2]. In 1964 Bell proposed an experiment for testing its existence in practice [3], which motivated the proposal of many similar types of experiments [4]. Such methods are nowcalled tests of Bell-type correlation. In 1981 the first experimental demonstration of such a test was done by Aspect, Grangier, and Roger in a photonic system, clearly establishing the existence of entanglement [5]. Today many experiments for such tests have been done in several types of physical systems, including photons [6], neutrons [7], trapped ions [8, 9], superconducting qubits [10], NV-centers in a diamond [11], and B mesons [12]. This has driven the recent growth of the field of quantum information, with quantum cryptography and quantum computation as good later examples. In 1984 and 1991, Bennett and Brassard, and Ekert independently proposed a cryptographic protocol using quantum systems [13, 14]. Surprisingly, the protocol is secure in principle for the ideal case, and the security of nonideal cases was proved later [15, 16]. In 1994 Shor proposed a computational algorithm for factorizing in prime integers using quantum systems [17]. It is again surprising that the protocol runs in polynomial time, something widely believed to be impossible for classical algorithms.With this cryptographic protocol and computational algorithmwhich outperform their classical counterparts as motivation, several newprotocols and algorithms have been proposed [18, 19], with experimentalists in diverse fields working to implement many of them in their labs [20, 21].